Determinism and Chaos
Published: June 23, 2011
Revised: September 6, 2013
The Second Law of Thermodynamics and Darwinian Evolution are
founded on arguments of identical form, the employment of which is
sufficient for drawing conclusions as to the gross behaviour of diverse
physical systems. These arguments are a part of the description of any
system which may be called deterministic; chaotic systems, in contrast,
are unpredictable precisely because they cannot be thus treated. The
chaotic behaviour of the general n-body problem, and the non-classical
features of Quantum Mechanics, follow from the impossibility of fulfilling
the prerequisites of determinism in the presence of 'sub-problems', and at
the level of elementary particles, respectively.
When a system is able to be described simultaneously at two different scales,
a probabilistic treatment of the assignment of small-scale states ('microstates')
to large-scale states ('macrostates') specifies the system's large-scale behaviour
for arbitrarily far in the future. Prediction is this act of specification, and a
predictable system is one that is deterministic. A deterministic system will
always occupy the macrostate with the greatest number of corresponding mi-
crostates — the macrostate's 'logical entropy' — which value is proportional to its
probalility of being realised.
2 Statistical Physics
Begin with an empty box divisible into two parts. Place some number of particles
in the box consecutively, assigning to each particle a random position and
zero speed. Take a given microstate of the system at any point in time to be a
specification of the position [and (null) velocity] of every particle of the frozen
gas, and a given macrostate to be one of three possibilities: either there are
particles only on one side; or there are particles only on the other side; or there
is at least one particle on each side. 1
1 Note that macrostates supervene on microstates.
The probability of occupying a given macrostate at some point in this additive
procedure is understood by examining a tree diagram which recognises that a
random arrangement of particles is the end of a process by which all of the
particles are introduced independently (see Figure 1). Each path through the
tree diagram is a 'possible history' of the construction of the system.
, P(0 : 2) = \
* P(l:l) = i
* P(l:l) = \
* P(2 : 0) = \
Figure 1: A tree diagram of the construction of a static and binary gas-in-a-box.
In general, the macrostate with both sides occupied is the most likely state
for the system to occupy, since it is in this macrostate that its logical entropy
is maximised. When the box contains two particles, there are twice as many
microstates corresponding to the macrostate in which both sides are occupied
as there are corresponding to each macrostate in which only one side is taken.
When the box contains many more than two particles, the maximum value of the
logical entropy is enormous compared to the alternatives. Indeed, the occupation
of the macrostate with the most microstates is both most likely and very likely:
this state is favoured over even the sum of the alternative states, of which there
can only ever be a few.
3 Natural Selection
Consider two species, A and B, of equal, large initial populations. Assume that
the ^4s are more fertile by a constant factor. The development of this system may
be partitioned into equally likely possible histories that specify precisely which
As and which Bs lived and died — into microstates thereby defined. A macrostate,
on the other hand, perhaps counts only the ratio of As and Bs alive after
each generation. For a random microstate, the corresponding macrostate will
probably be one in which the As have been 'naturally selected' for their superior
reproductive ability. After many generations, the ^4s will [almost certainly] vastly
outnumber the Bs, this eventuality corresponding to the Evolution of a higher
birth rate in the common ancestor of these two species.
A state of a system in which Evolution occurs is a state to which the principle
of Natural Selection applies, and the state to which that system evolves is the
one in which its logical entropy is maximised. Evolution may be driven by any
sort of microdynamics, these the actual physics by which a microstate is chosen.
One species may be more fertile or superior in combat; there may be competition
. \x Particle
' - - - Left
for resources. It may be an intricate mixture of factors which induces Evolution.
The only necessity is that one [be able to] choose the microstates of the system
such that each is equally likely. Natural Selection depends not at all on how
changes in phenotype arise; it requires only descent with random modification.
The development of a possible history for the action of some Natural Selection
is analogous to the construction of the static gas-in-a-box by the placement of each
particle independent of the others. In the former case, the random modification
of the system is a small change in phenotype; in the latter, it is the addition of
a particle to the statistical system. One may construct a tree diagram for the
Evolution of individual biologic traits, and one may describe each column of any
tree diagram as 'descended' from the previous columns. Indeed, the construction
of the static gas-in-a-box may be understood as obeying Natural Selection, with
a gaseous macrostate of a higher logical entropy being more 'fit' to have survived
the perturbations applied to previous configurations of that same macrostate; as
particles are added to the box, the asymmetric macrostates 'die out', while the
remaining possibility persists until it is the only one left.
The Principle of Indifference says that for mutually exclusive, jointly exhaustive,
and indistinguishable possibilities, the probabilities of each being realised are
equal in magnitude. If the possibilities are indistinguishable, then none could be
preferred. More precisely, the different possibilities must be indistinguishable
in the sense that whatever distinguishes them should not affect their relative
likelihood; but they must be able to be identified as distinct. The non-dynamical
features ('labels') that do distinguish equally likely possibilities are said to be
on a scale smaller than that of the system itself.
In creating a model for a physical system, adopting simplifying assumptions,
one assigns both the possibilities and the labels which differentiate them, so a
difference in scale is a specification of scope: something modelled is 'small' when
one is justified in not considering it, and in not considering something, that
thing is not included in whatever model is being proposed. Point particles are
precisely those dynamical objects which have random interactions. These objects
with no [significant] internal structure, then, are the most basic constituents of
every deterministic system.
The Fundamental Postulate of Thermodynamics did not need to be considered
explicitly in the above generalisation of the Second Law, because in a dynamical
gas-in-a-box (i.e. one in which the velocities of the particles are non-trivial) the
temporal evolution of the system is determined by the speed of particles and
the time interval for which the system is observed. Such a gaseous system then
evolves toward a state characterised by an even distribution of particles, and
the state of the evolution toward the state with the maximum logical entropy is
itself a (steady) [macro]state with a maximised logical entropy.
5 Determinism and Chaos
The two-body problem, being solvable, is an example of a deterministic system,
and it is solvable because it is elementary: it contains no sub-problems — that is,
internal systems which are themselves problems on the scale of the system itself.
In the case of the n-body problem, a sub-problem may be that same physical
system constructed in terms of groups of those n bodies. Such a construction is
possible for three or more bodies, but impossible for two: if one of the bodies
in the two-body problem is removed, the system is then indescribable (and
in no way a 'problem'). The two-body problem is solvable because it is the
simplest n-body problem and so contains no sub-problems. On the other hand,
the three-body problem, for instance, does contain a sub-problem (namely the
two-body problem) and so is not deterministic, but instead chaotic, in behaviour.
Whereas any system is, at any time, in one of an enumerable set of possible
states, with a chaotic system specifically this set has more than one member;
a chaotic system is chaotic as it oscillates [unpredictably] among the multiple
possibilities. With determinism, in contrast, there is only one state [likely] to
be occupied. Consider a spinning coin in a coin toss, another chaotic system.
Here there are two states, 'heads up' and 'heads down', and the coin may indeed
be said to occupy one or the other at any given point in the toss; if there were
instead just one state, this would provide a stable equilibrium for the coin, such
as exists for a spinning sphere.
Sub-problems involve chaotic behaviour because they are interdependent;
with sub-problems, no logical entropy can be defined, because one can con-
struct no formalism which directly partitions possible histories into [equally
likely] microstatcs. Indeed, a tree diagram cannot describe the dynamics of the
sub-problems, the existence of which prevents the choice of path through the
tree from being random. If and only if an appropriate logical entropy could be
constructed would there be a [single] state with the greatest logical entropy, and
the future of the system could then be [deterministically] predicted [to be an
occupation of that (stable) state] .
6 Quantum Mechanics
Now assume that a given system is composed of 'elementary particles' [and
that the system itself is on the scale of those particles] . An elementary particle
here is one that is indivisible in the sense that, when it is 'divided', it yields
constituents that are on the same scale as the original. Elementary particles are
indistinguishable because they cannot be marked by labels as can, for example,
the two sides of a coin.
The 'peculiar' form of the laws of Quantum Mechanics is a logical consequence
of the presence of elementary particles, the scale of which is the domain of
the physical theory. To observe an elementary particle [directly] is to learn the
dynamics of the absolute smallest constituents of a small-scale system and to be
able to incorporate them in the model used. These dynamics then cannot relate
well-defined microstates, because, to be microdynamics, they must be unknown.
With observation at the smallest scale, then, the system itself changes, for there
no longer can be any microdynamics at that scale, and the logic of determinism
no longer applies. Then the dynamics of the observed particles are the only
dynamics left to be seen. 2
2 In the case of the double-slit experiment, the determinism prohibited by observation is
what produces the large-scale interference pattern visible on the screen when the system is
It is not just sufficient that systems be large to behave classically; it is
necessary, too. 3 Quantum Mechanics is then not a break with the principles
of Classical Mechanics; rather it is the consequence of them, these principles
being the very ones which allow for the classical treatment of, e.g. , the two-body
problem. Determinism is fundamental, and it manifests differently at and away
from the minimum scale.
As Quantum Mechanics gives a probabilistic result that a subatomic particle will
be in this or that state, so does a model of a spinning coin yield a probability for
each possible state of the coin at any time (these probabilities being decided by the
coin's fairness). In Quantum Mechanics, the collapse of a wave-function places the
system in one or the other state, as a coin's position function 'collapses' to either
'heads up' or 'heads down' when it is observed. In both cases, if the system
then goes unobserved for sufficient time, the object regains its unpredictability.
The three-body problem, too, behaves in this fashion, with its oscillations being
between states of high and low potential energy. The quantisation of states in
Quantum Mechanics is part of the multiplication of states in chaos; precisely
with determinism is there ever only one possible smooth end-state.
The difference between quantum and classical chaos is that in the former it is
the minimum scale of the system that is the cause of the unpredictable behaviour,
while in the latter it is the existence of sub-problems (as in the three-body
problem) or the flatness of the coin (as in the coin toss) . Determinism always
fails to describe a system because of properties of that system which prohibit
the construction of microstates and macrostates in the required fashion. A
probability-based determination of the state of a system is, in this sense, the
default determination of all dynamics. Only when a system transcends spatial
and temporal scales may one recognise one possibility as favoured. For instance,
the systems of Natural Selection and Statistical Mechanics transcend scales; but
those of Quantum Mechanics and Chaos Theory do not. Notably, these last
three disciplines are just those subjects in theoretical physics in which probability
plays a foundational role.
Appendix A E.T. Jaynes
The author here recognises a number of similarities between his own thought
and that of E.T. Jaynes, whose work he first encountered only as this article was
nearing completion. The reader is referred to Jaynes's corpus, and specifically to
the works cited in the bibliography, for the purpose of comparison and especially
if he seeks a mathematical treatment of the common notions.
 E.T. Jaynes. Prior Information in Inference. 1982.
3 Black holes are exceptions: like the elementary particles of the smallest scale, they have
no internal structure and so, too, no microdynamics.
 E.T. Jayncs. Macroscopic prediction. In H. Haken, editor, Complex Systems
- Operational Approaches, page 254. Springer- Verlag, Berlin, 1985.
 E.T. Jaynes. How Should We Use Entropy in Economics? doi:10.1. 1.41.4606,
 E.T. Jaynes. The Second Law as Physical Fact and Human Inference.
doi:10. 1.1. 77.7388, 1998.